Solve frac{5sqrt{2}+3sqrt{3}}{3sqrt{2}+2sqrt{3}}

$$\frac { 5 \sqrt { 2 } + 3 \sqrt { 3 } } { 3 \sqrt { 2 } + 2 \sqrt { 3 } }$$

Evaluate

$$-\frac{\sqrt{6}}{6}+2\approx 1.59175171$$

Solution Steps

Rationalize the denominator of \frac{5\sqrt{2}+3\sqrt{3}}{3\sqrt{2}+2\sqrt{3}} by multiplying numerator and denominator by 3\sqrt{2}-2\sqrt{3}.

$$\frac{\left(5\sqrt{2}+3\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)}{\left(3\sqrt{2}+2\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)}$$

Consider \left(3\sqrt{2}+2\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(5\sqrt{2}+3\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)}{\left(3\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}}$$

Expand \left(3\sqrt{2}\right)^{2}.

$$\frac{\left(5\sqrt{2}+3\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)}{3^{2}\left(\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(5\sqrt{2}+3\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)}{9\left(\sqrt{2}\right)^{2}-\left(2\sqrt{3}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(5\sqrt{2}+3\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)}{9\times 2-\left(2\sqrt{3}\right)^{2}}$$

Multiply 9 and 2 to get 18.

$$\frac{\left(5\sqrt{2}+3\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)}{18-\left(2\sqrt{3}\right)^{2}}$$

Expand \left(2\sqrt{3}\right)^{2}.

$$\frac{\left(5\sqrt{2}+3\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)}{18-2^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(5\sqrt{2}+3\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)}{18-4\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(5\sqrt{2}+3\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)}{18-4\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{\left(5\sqrt{2}+3\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)}{18-12}$$

Subtract 12 from 18 to get 6.

$$\frac{\left(5\sqrt{2}+3\sqrt{3}\right)\left(3\sqrt{2}-2\sqrt{3}\right)}{6}$$

Apply the distributive property by multiplying each term of 5\sqrt{2}+3\sqrt{3} by each term of 3\sqrt{2}-2\sqrt{3}.

$$\frac{15\left(\sqrt{2}\right)^{2}-10\sqrt{3}\sqrt{2}+9\sqrt{3}\sqrt{2}-6\left(\sqrt{3}\right)^{2}}{6}$$

The square of \sqrt{2} is 2.

$$\frac{15\times 2-10\sqrt{3}\sqrt{2}+9\sqrt{3}\sqrt{2}-6\left(\sqrt{3}\right)^{2}}{6}$$

Multiply 15 and 2 to get 30.

$$\frac{30-10\sqrt{3}\sqrt{2}+9\sqrt{3}\sqrt{2}-6\left(\sqrt{3}\right)^{2}}{6}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{30-10\sqrt{6}+9\sqrt{3}\sqrt{2}-6\left(\sqrt{3}\right)^{2}}{6}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{30-10\sqrt{6}+9\sqrt{6}-6\left(\sqrt{3}\right)^{2}}{6}$$

Combine -10\sqrt{6} and 9\sqrt{6} to get -\sqrt{6}.

$$\frac{30-\sqrt{6}-6\left(\sqrt{3}\right)^{2}}{6}$$

The square of \sqrt{3} is 3.

$$\frac{30-\sqrt{6}-6\times 3}{6}$$

Multiply -6 and 3 to get -18.

$$\frac{30-\sqrt{6}-18}{6}$$

Subtract 18 from 30 to get 12.

$$\frac{12-\sqrt{6}}{6}$$